If a neural network is a universal function approximator, can it have any prior beliefs?
I think your deduction is mostly correct.

Neural networks of depth are universal function approximators. This means that in principal, for any function of the form you describe, there's a NN that approximates it. 

However, a particular NN architecture of fixed width and depth, with fixed connections is not a universal approximator for all functions. Only an infinitely wide NN is, and that's a theoretical construct, not something you can make in practice.

Typically, a practitioner using a NN injects their prior beliefs about the problem  by selecting an architecture. For example, it is common to use the ImageNet or ResNet architectures for image processing tasks. Those architectures are less effective on other types of tasks. For instance, it should be clear that they are ineffective on say, a time-series analysis task.